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Chapter 7: Problem 97

Evaluate: \(8 \div \frac{1}{2}\)

Short Answer

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Step by step solution

01

Express the Division as Multiplication

To divide by a fraction, multiply by its reciprocal. The reciprocal of \(\frac{1}{2}\) is \(\frac{2}{1} \) or simply 2. Thus, \( 8 \div \frac{1}{2} \) can be written as \( 8 \times 2 \).
02

Perform the Multiplication

Now, multiply 8 by 2. \( 8 \times 2 = 16 \).
03

Write the Final Answer

Therefore, \( 8 \div \frac{1}{2} = 16 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

reciprocal
In mathematics, the concept of the reciprocal is fundamental when working with fractions. To find the reciprocal of a fraction, you simply swap its numerator and denominator.

For example, the reciprocal of \(\frac{1}{2}\) is \(\frac{2}{1}\)\ , or simply 2. Essentially, you are flipping the fraction upside down. The reciprocal of a number is also known as its multiplicative inverse because multiplying a number by its reciprocal always gives 1.

Consider the fraction \(\frac{3}{4}\):
  • The reciprocal is \(\frac{4}{3}\).

When dealing with whole numbers such as 5, the reciprocal is \(\frac{1}{5}\). This principle is crucial for understanding how to reverse operations, like division with fractions.
multiplication
Multiplication is one of the four basic arithmetic operations. When we talk about multiplying fractions, the process is straightforward: Simply multiply the numerators together and then the denominators together.

Let's break it down:
  • Take the fractions \(\frac{a}{b} \times \frac{c}{d}\).
  • Multiply the numerators: \(a \times c\).
  • Multiply the denominators: \(b \times d\).

For example, if you have \(\frac{2}{3} \times \frac{4}{5}\), you would calculate:
  • Numerator: 2 \times 4 = 8
  • Denominator: 3 \times 5 = 15

So, \(\frac{2}{3} \times \frac{4}{5} = \frac{8}{15}\).

Remember, the key idea is multiplying the top numbers (numerators) together and the bottom numbers (denominators) together. This simplicity is why multiplication often replaces division when handling fractions.
fraction division
Fraction division might seem complex at first, but it becomes straightforward when you understand the steps involved. To divide by a fraction, you multiply by its reciprocal.

Here鈥檚 a step-by-step breakdown:
  • Step 1: Find the reciprocal of the divisor (the fraction you are dividing by).
  • Step 2: Change the division operation to multiplication.
  • Step 3: Multiply as you would with regular fractions.

Consider the exercise: \(8 \div \frac{1}{2}\).

Step 1: Find the reciprocal of \(\frac{1}{2}\), which is 2.

Step 2: Change the division sign to multiplication: \(8 \times 2\).

Step 3: Multiply: 8 \times 2 = 16.

By converting the division into a multiplication problem, we simplify the process.
This method is useful for all fractions. Just remember: Division by a fraction is the same as multiplication by its reciprocal.

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Most popular questions from this chapter

For exercises 1-10, (a) solve. (b) check. $$ \frac{4}{9} p-\frac{1}{8}=\frac{25}{72} $$

For exercises 11-30, (a) solve. (b) check. $$ \frac{13}{d}-\frac{5}{9}=\frac{1}{6} $$

For exercises \(41-44\), the formula \(R=\frac{V C}{T}\) describes the flow rate of fluid \(R\) through an intravenous drip. Is the relationship of the given variables a direct variation or an inverse variation? $$ V \text { and } C \text { are constant; the relationship of } R \text { and } T \text {. } $$

For exercises 11-30, (a) solve. (b) check. $$ \frac{3}{10}+\frac{7}{m}=\frac{14}{m}+\frac{1}{15} $$

$$ \text { Solve: } 0.75=\frac{k}{60} $$
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